
in terms of coordinates \(x^{M} \equiv (x^{\mu },y)\) , where \(y \in (-\pi r_{c},+\pi r_{c}]\)  parameterizes an orbifolded extra-dimensionThat is, the extra-dimension is a circle in which the points with coordinates \(y\)  and \(-y\)  are identified. As we will see, this view of the extra dimension (as opposed to treating it as a line-segment) allows us to motivate and use the boundary conditions of the Kaluza-Klein mode equations at the orbifold fixed-points at \(y=0\)  and \(y=\pi \)  more easily.; for convenience we define \(\varphi \equiv y/r_{c}\) , and use \(y\)  or \(\varphi \)  interchangeably as the coordinate of the fifth dimension. Note that this form of the metric was used in our previous works [1]}, [2]}, [3]} for the unstabilized RS1 metric, following [4]} to bring the quadratic Lagrangian to a canonical form from the outset. In the case of the stabilized model, the situation is further complicated by a nontrivial mixing between the bulk scalar field and the scalar metric fluctuations. As we will explain below and demonstrate explicitly in Appendix , the total derivative \(\Delta \mathcal {L}\)  helps us bring the Lagrangian into a canonical form.
